{"id":257234,"date":"2026-07-13T15:23:29","date_gmt":"2026-07-13T12:23:29","guid":{"rendered":"https:\/\/1kitap1.com\/en\/cryptographic-algorithms-anita-tomar-ankur-nehra\/"},"modified":"2026-07-13T15:23:29","modified_gmt":"2026-07-13T12:23:29","slug":"cryptographic-algorithms-anita-tomar-ankur-nehra","status":"publish","type":"post","link":"https:\/\/1kitap1.com\/en\/cryptographic-algorithms-anita-tomar-ankur-nehra\/","title":{"rendered":"Cryptographic Algorithms – Anita Tomar Ankur Nehra"},"content":{"rendered":"
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where x is the quadratic residue modulo p. The answer to the above problem can be determined from the following result: 4. Euler\u2019s Criterion: x is a quadratic residue modulo p if and only if x p \u22121 2 \u22611 \u00f0mod p\u00de. Remark 3.7: Suppose z is a quadratic residue and p \u22613 \u00f0mod 4\u00de. Then, the two square roots of z modulo p are \u00b1 z p + 1 4 \u00f0mod p\u00de. Example 3.8: Let E be an elliptic curve E: y2 = x3 + x + 6 defined over Z11 with a = 1, b = 6, and p = 11.<\/p>\n

For each value x 2 Z11, we compute x3 + x + 6 \u00f0mod 11\u00de as follows: Thus, E has 13 points, including O. If we take two distinct points P = \u00f05, 2\u00de and Q = \u00f02, 7\u00de, then their sum R = P + Q = \u00f0x3, y3\u00de can be explained as: \u03bb = 7 \u22122 2 \u22125 = 5 \u22123 \u226116 8 = 2 \u00f0mod 11\u00de x3 = 22 \u22125 \u22122 = \u22123 \u22618 \u00f0mod 11\u00de and y3 = 2\u00f05 \u22128\u00de \u22122 = \u22128 \u22613 \u00f0mod 11\u00de Hence, R = P + Q = \u00f08, 3\u00de.<\/p>\n

A further doubling P = \u00f05, 2\u00de can be described as: \u03bb = 3\u00f05\u00de2 + 1 2 \u00d7 2 = 76 4 \u226119 \u22618 \u00f0mod 11\u00de x3 = 82 \u22122 \u00d7 5 = 54 \u226110 \u00f0mod 11\u00de Table 3.1: Calculation of points on the elliptic curve E : y2 = x3 + x + 6 \u00f0mod 11\u00de. x3 + x + 6 \u00f0mod 11\u00de Quadratic residue y No No Yes Yes No Yes No Yes Yes No Yes 3.4 Mathematical Background of Elliptic Curve Cryptography and y3 = 8\u00f05 \u221210\u00de \u22122 = \u221242 \u22612 \u00f0mod 11\u00de hence R = P + P = 2P = \u00f010, 2\u00de.<\/p>\n

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e-ISBN (PDF) !-#-$$-$#$ %3-! e-ISBN (EPUB) !-#-$$-$#$ %- Anita Tomar, Ankur Nehra Cryptographic Algorithms Elliptic and Jacobian, Elliptic Curve Cryptography and Computational Security Authors Prof. Anita Tomar Department of Mathematics Sri Dev Suman Uttarakhand University Pt. L.M.S. Campus Rishikesh 249201, Uttarakhand, India anitatmr@yahoo.com Dr. Ankur Nehra Department of Mathematics Dhanauri P.G.<\/p>\n

College Dhanauri 247667, Haridwar, Uttarakhand, India ankurnehra123@gmail.com ISBN 978-3-11-914287-8 e-ISBN (PDF) 978-3-11-222160-0 e-ISBN (EPUB) 978-3-11-222190-7 Library of Congress Control Number: 2025946535 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the internet at http:\/\/dnb.dnb.de. \u00a9 2026 Walter de Gruyter GmbH, Berlin\/Boston, Genthiner Stra\u00dfe 13, 10785 Berlin Cover image: ArtemisDiana\/iStock\/Getty Images Plus Typesetting: Integra Software Services Pvt.<\/p>\n

Ltd. Printing and binding: CPI books GmbH, Leck www.degruyterbrill.com Questions about General Product Safety Regulation: productsafety@degruyterbrill.com \u201cDedicated to beloved parents and family members\u201d for their love, endless support, encouragement and sacrifices Preface The rich mathematical heritage of ancient India has captivated scholars and enthusi\u00ad asts for centuries. As our world increasingly depends on secure digital communica\u00ad tion and sophisticated information systems, the need for advanced cryptographic techniques has never been more urgent. In this context, there is a growing recogni\u00ad tion of the profound insights that ancient Indian mathematics (AIM) can offer to the field of cryptography.<\/p>\n

This book, Cryptographic Algorithms: Elliptic and Jacobian, Elliptic Curve Cryptog\u00ad raphy and Computational Security, embarks on a fascinating journey at the confluence of historical wisdom and modern technology. It illustrates how ancient Indian mathe\u00ad matical principles can transform contemporary cryptographic practices, bridging mil\u00ad lennia of mathematical evolution with the forefront of digital security. We aim to provide a comprehensive reference for researchers, engineers, practi\u00ad tioners, and students who are eager to explore the deep connections between the timeless principles of AIM and the rapidly advancing field of modern cryptography.<\/p>\n

By exploring foundational concepts, algorithms, and practical applications that unite these two domains, this book seeks to rekindle appreciation for the enduring bril\u00ad liance of ancient Indian mathematical genius and its role in shaping future secure dig\u00ad ital ecosystems. The foundation of this work is a thorough exploration of classical and state-of-the- art cryptographic concepts.<\/p>\n<\/blockquote>\n

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.<\/em><\/p>\n

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